Why does my model tear apart at its vertices when attempting to pose?
As indicated in my comment, your problem seems to be loose geometry.  I found I could clean it up fairly easily by selecting all vertices of your mesh in Edit mode, then choosing Mesh-->Clean Up-->Remove Doubles and specifying a "Merge Distance" of .06.  I can't tell whether that took out details of your mesh that you wanted to keep, but this resulted in a mesh without loose parts.  Re-parenting the mesh to the armature with automatic weights made everything work as expected.  You could try playing with the Merge Distance of the "Remove Doubles" command if parts of your mesh are getting oversimplified.